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July 2019, 39(7): 3867-3895. doi: 10.3934/dcds.2019156

Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $ \mathbb{R}^+ $

1. 

Massachusetts Institute of Technology, Cambridge, MA 02139, USA

2. 

University of Illinois, Urbana-Champaign, IL 61801, USA

* Corresponding author: N. Tzirakis

Received  April 2018 Revised  January 2019 Published  April 2019

Fund Project: E.C. was supported by a National Physical Science Consortium fellowship and by NSF MSPRF #1704865. N.T. was supported by a grant from the Simons Foundation (#355523 Nikolaos Tzirakis) and by Illinois Research Board, RB18051

In this paper we establish an almost optimal well-posedness and regularity theory for the Klein-Gordon-Schrödinger system on the half line. In particular we prove local-in-time well-posedness for rough initial data in Sobolev spaces of negative indices. Our results are consistent with the sharp well-posedness results that exist in the full line case and in this sense appear to be sharp. Finally we prove a global well-posedness result by combining the $L^2$ conservation law of the Schrödinger part with a careful iteration of the rough wave part in lower order Sobolev norms in the spirit of the work in [5].

Citation: E. Compaan, N. Tzirakis. Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $ \mathbb{R}^+ $. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3867-3895. doi: 10.3934/dcds.2019156
References:
[1]

I. BejenaruS. HerrJ. Holmer and D. Tataru, On the 2D Zakharov system with $L^2$ Schrödinger data, Nonlinearity, 22 (2009), 1063-1089. doi: 10.1088/0951-7715/22/5/007.

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J. L. Bona, S. M. Sun and B. Y. Zhang, Nonhomogeneous boundary-value problems for one-dimensional nonlinear Schrödinger equations, J. Math. Pures Appl. (9), 109 (2018), 1–66, arXiv: 1503.00065. doi: 10.1016/j.matpur.2017.11.001.

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J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part Ⅰ: Schrödinger equations, GAFA, 3 (1993), 107-156. doi: 10.1007/BF01896020.

[4]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part Ⅱ: The KdV equation, GAFA, 3 (1993), 209-262. doi: 10.1007/BF01895688.

[5]

J. CollianderJ. Holmer and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Trans. Amer. Math. Soc., 360 (2008), 4619-4638. doi: 10.1090/S0002-9947-08-04295-5.

[6]

J. E. Colliander and C. E. Kenig, The generalized Korteweg-de Vries equation on the half line, Comm. Partial Diff. Equations, 27 (2002), 2187-2266. doi: 10.1081/PDE-120016157.

[7]

E. Compaan and N. Tzirakis, Well-posedness and nonlinear smoothing for the "good" Boussinesq equation on the half-line, J. Differential Equations, 262 (2017), 5824-5859. doi: 10.1016/j.jde.2017.02.016.

[8]

M. B. Erdoǧan and N. Tzirakis, Regularity properties of the cubic nonlinear Schrödinger equation on the half line, J. Funct. Anal., 271 (2016), 2539-2568. doi: 10.1016/j.jfa.2016.08.012.

[9]

M. B. Erdoǧan and N. Tzirakis, Regularity properties of the Zakharov system on the half line, Comm. Partial Differential Equations, 42 (2017), 1121-1149. doi: 10.1080/03605302.2017.1335320.

[10]

J. GinibreY. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436. doi: 10.1006/jfan.1997.3148.

[11]

H. Pecher, Low regularity well-posedness for the 3D Klein-Gordon-Schrödinger system, Commun. Pure Appl. Anal., 11 (2012), 1081-1096. doi: 10.3934/cpaa.2012.11.1081.

[12]

H. Pecher, Some new well-posedness results for the Klein-Gordon-Schrödinger system, Differential Integral Equations, 25 (2012), 117-142.

show all references

References:
[1]

I. BejenaruS. HerrJ. Holmer and D. Tataru, On the 2D Zakharov system with $L^2$ Schrödinger data, Nonlinearity, 22 (2009), 1063-1089. doi: 10.1088/0951-7715/22/5/007.

[2]

J. L. Bona, S. M. Sun and B. Y. Zhang, Nonhomogeneous boundary-value problems for one-dimensional nonlinear Schrödinger equations, J. Math. Pures Appl. (9), 109 (2018), 1–66, arXiv: 1503.00065. doi: 10.1016/j.matpur.2017.11.001.

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part Ⅰ: Schrödinger equations, GAFA, 3 (1993), 107-156. doi: 10.1007/BF01896020.

[4]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part Ⅱ: The KdV equation, GAFA, 3 (1993), 209-262. doi: 10.1007/BF01895688.

[5]

J. CollianderJ. Holmer and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Trans. Amer. Math. Soc., 360 (2008), 4619-4638. doi: 10.1090/S0002-9947-08-04295-5.

[6]

J. E. Colliander and C. E. Kenig, The generalized Korteweg-de Vries equation on the half line, Comm. Partial Diff. Equations, 27 (2002), 2187-2266. doi: 10.1081/PDE-120016157.

[7]

E. Compaan and N. Tzirakis, Well-posedness and nonlinear smoothing for the "good" Boussinesq equation on the half-line, J. Differential Equations, 262 (2017), 5824-5859. doi: 10.1016/j.jde.2017.02.016.

[8]

M. B. Erdoǧan and N. Tzirakis, Regularity properties of the cubic nonlinear Schrödinger equation on the half line, J. Funct. Anal., 271 (2016), 2539-2568. doi: 10.1016/j.jfa.2016.08.012.

[9]

M. B. Erdoǧan and N. Tzirakis, Regularity properties of the Zakharov system on the half line, Comm. Partial Differential Equations, 42 (2017), 1121-1149. doi: 10.1080/03605302.2017.1335320.

[10]

J. GinibreY. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436. doi: 10.1006/jfan.1997.3148.

[11]

H. Pecher, Low regularity well-posedness for the 3D Klein-Gordon-Schrödinger system, Commun. Pure Appl. Anal., 11 (2012), 1081-1096. doi: 10.3934/cpaa.2012.11.1081.

[12]

H. Pecher, Some new well-posedness results for the Klein-Gordon-Schrödinger system, Differential Integral Equations, 25 (2012), 117-142.

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